多分散有序盘组中的几何阻挫和颗粒间隙尺寸分布
Geometric Frustration and Interparticle Gap Size Distributions in Ordered Hexagonal Polydisperse Disk Packs
摘要：
这项工作分析了在xy平面中的六边形单元结构的系综中产生的颗粒间间隙的分布和尺寸，当填充具有平均值（r）和标准偏差$Δr$的半径的高斯分布的盘时。在这个调查的过程中，一个等效的间隙之间产生的六边形单元结构包和九球台球架模式。导出了$Γ$量级粒子间能隙的概率分布和位置的解析表达式。由于变量的数量和大量的可能的安排，蒙特卡罗模拟已经进行了补充和探索的解析形式为三个非常不同的系统：i）台球与美国台球大会（BCA）的规格，ii）美国便士与美国铸币局的规格，和iii）一个假设的系统与$r = 1.0 m$和$Δr = 1x10^{-10}$ m对应的一个原子半径的规模。在每种情况下，分别计算了上述那些$Δr$以及2 $Δr$和0.5 $Δr$的间隙尺寸的概率密度分布。一个一般的结果是一个非零归一化（$\fracΓ{Δr}$）的间隙大小出现的概率，$P(Γ\geq αΔr)= 1-0.124α$，其中$α$是一个常数$\leq 5.0$。这个奇怪的结果反映了几何挫折的现象，系统无法同时满足所有的几何约束所需的一个完美的机架sans粒子间的差距。
英文摘要：
This work analyzes the distribution and size of interparticle gaps arising in an ensemble of hexagonal unit structures in the xy plane when packing disks with a Gaussian distribution of radii with mean (r) and standard deviation $Δr$. During the course of this investigation an equivalency is established between gaps arising in hexagonal unit structure packs and nine-ball billiard rack patterns. An analytic expression is derived for the probability distribution and location of interparticle gaps of magnitude $Γ$. Due to the number of variables and large number of possible arrangements, a Monte Carlo simulation has been conducted to complement and probe the analytic form for three very different systems: i) billiard balls with Billiard Congress of America (BCA) specifications, ii) US pennies with specifications of the US Mint, and iii) a hypothetical system with $r = 1.0 m$ and $Δr = 1x10^{-10}$ m corresponding to the scale of one atomic radius. In each case, probability density distributions of gap sizes have been calculated for those $Δr$ above, and also for 2$Δr$ and 0.5$Δr$, respectively. A general result is presented for the probability of a nonzero normalized ($\fracΓ{Δr}$) gap size arising, $P(Γ\geq αΔr)= 1-0.124α$, where $α$ is a constant $\leq 5.0$. This curious result reflects the phenomenon of geometric frustration; the inability of the system to simultaneously satisfy all geometric constraints required by a perfect-rack sans interparticle gaps.
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